Vacuum neutrino oscillations with relativistic wave ... · Vacuumneutrinooscillationswithrelativisticwavepacketsinquantumfieldtheory Introduction Ourapproach:QFTwithrelativisticwavepackets

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theory

Vacuum neutrino oscillations with relativistic wavepackets in quantum field theory

Dmitry V.Naumov

JINR, Dubna, Russia

November 21, 2011

Based uponD. V. Naumov and V. A. Naumov, J. Phys. G 37 (2010) 105014V. A. Naumov and D. V. Naumov, Russian Phys. J. 53, 6/1 (2010) 5D. V. Naumov, V. A. Naumov, [arXiv:1110.0989 [hep-ph]].

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryIntroductionWhat is wrong with QM theory of neutrino oscillations?

Quantum-mechanical theory of neutrino oscillations is used to analyzethe neutrino experimental data. However QM theory is contradictoryand not complete. Used assumptions are questionable:

4 Why coherent state |να〉 =∑i Vαi|νi〉 can be produced in a reaction with `αwhile a state |`i〉 =∑

α V∗αi|α〉 is likely to be incoherent?4 Same pν of all massive |νi〉 is unphysical as it depends on reference frame.4 Definite momentum pν implies uncertain position of neutrino δXν ∝ ∞4 Ultra-relativistic assumption is reference frame dependent and sometimeserroneously lead to factor two difference in oscillation phase

The origin of problems of QM approach is due to ignoring of productionand detection processes. Thus QM postulates the wave function ofneutrino and this is the source of numerous paradoxes.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryIntroductionWhat is wrong with QM theory of neutrino oscillations?

What can be found in the literature beyond naive QM approach?Neutrino is considered as wave packets either in QM or in QFT

This successfully resolves a number of problems of naive theory. But suchan approach has its own problems:

. the form of wave packet is a guess. It is hard to quantify the ”width” ofneutrino wave packets

. the wave packets used in the literature are essentially non-relativistic whichis far from (almost) any experiment with neutrino.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryIntroductionOur approach: QFT with relativistic wave packets

Creation and annilihation of particles is described in a consistent way inQFTStandard S-matrix theory of QFT works with states of definitemomentum=states uniformly distributed over all space. This is a goodapproximation for microscopic scales. It is not valid for calculations ofprocesses localized in space and time and separated by macroscopicintervals.To describe appropriate states one needs a theory of relativistic wavepackets. We constructed such a theoryRelativistic wave packets (RWP) — states described by meanmomentum and coordinate and their corresponding «widths»The mean position of RWP follows classical trajectoryA probability of collision of several RWP is given by∫

d4x∏in,out

|wave functionin(x− xin)|2|wave functionout(x− xin)|2

Our formalism generalizes for 4-D space the impact parameter whichnaturally describes suppression of collision probability with notoverlaping RWPs

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QFT + localized in space-time relativistic wavepackets in (source and detector)Macroscopic Feynman diagram (source and detectorcan be separated by thousands of km)Neutrino is a virtual particle (NO hypothesis about its4-momentum, 4-coordinate, etc)Neutrino ”oscillations” are not mutual transformationsνα ↔ νβ , but just a result of interference of diagramswith virtual massive neutrino νi as well as”oscillations” of neutral kaons, D mesons and othersimilar known systems!

- Grimus, Stockinger 1996 Phys. Rev. D54 3414 (arXiv:hep-ph/9603430)- Cardall 2000 Phys. Rev. D 61 073006(arXiv:hep-ph/9909332)- Beuthe 2003 Phys. Rept. 375 105(arXiv:hep-ph/0109119)- D. V. Naumov and V. A. Naumov, J.Phys. G 37 (2010) 105014- V. A. Naumov and D. V. Naumov,Russian Phys. J. 53, 6/1 (2010) 5

Both diagrams are possible in the SM!L = −

∑i,α

g√2VαilαLγµνiLWµ

+ h.c

i

ℓ

ℓ

i

j

W W W W

. . . . . .


An outlook on obtained results without equations• Instead of postulating the neutrino wave function as done in QMapproach, we calculated neutrino wave function — found to be(dispersing in space-time ) RWP.

• Number of events of a macroscopic process is factorized intoΦ× Pαβ × σ

• Standard QM formula for ν-oscillations — is an approximation of amore general formula which depends. time intervals of neutrino source and detector (relevant for modernaccelerator expeiments);

. type of reaction of neutrino production and detection, its kinematics;

. dimension of source and detector and distance between them.• Our formula contains suppression of interference at distances largerthan coherence length as well for incoherent superposition of states.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryShortly about wavepackets and S-matrixWave packet state in momentum and coordinate spaces, integrals of motion

Fock state:|k, s〉 =√2Ek a†ks|0〉,

Ek =

√k2 + m2

Singular normalization:〈q, r|k, s〉 = (2π)32Ekδsrδ (k− q).

Conventional commutation relationsaqr, aks = a†qr, a†ks = 0,

aqr, a†ks = (2π)3δsrδ (k− q).

Wave packet

|p, s, x〉 =∫ dkφ(k, p)ei(k−p)x

(2π)32Ek|k, s〉,

φ(k, p) is Lorentz invariant, with a sharp peakat k = p and a width σNon singular normalization:

〈q, r, y|p, s, x〉 = δsrei(qy−px)D(p, q; x− y),with relativistic-invariant functionD(p, q; x) = ∫ dk

(2π)32Ek φ(k, p)φ∗(k, q)eikx

φ(k, p) can be thought as Fourie of the “sourcefunction“ j(x) in (i∂ − m)ψ(x) = j(x)

Both Fock and Wave packet states transform in the same way under the Lorentz transfor-mations.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryShortly about wavepackets and S-matrixWave packet state in momentum and coordinate spaces, integrals of motion..

The normalization of the WP state is finite:〈p, s, x|p, s, x〉 = D(p, p; 0) = 2EpV(p). (1)

The quantities Ep and V(p) in (1) are, respectively, the mean energy andeffective spatial volume of the packet, defined by

Ep =∫ dxψ(p, x)i∂0ψ∗(p, x)∫ dx|ψ(p, x)|2 =

1

V(p)∫ dk|φ(k, p)|2

4(2π)3Ek , (2)

V(p) =∫dx|ψ(p, x)|2 =

∫ dk(2π)3

|φ(k, p)|2(2Ek)2 =

V(0)Γp

, (3)

where Γp = Ep/m. So, both Ep and V(p), as well as the mean momentum pdefined by a relation similar to (2), are integrals of motion.The mean position of the packet follows the classical trajectory:

x =1

V(p)∫dxψ∗(p, x)xψ(p, x) = vpx0. (4)

Here vp = p/Ep is the mean group velocity of the packet, which coincideswith the most probable velocity ∇pEp = p/Ep.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryShortly about wavepackets and S-matrixRelativistic Gaussian packets (RGP).

assume a simple form for:

φ(k, p) = 2π2

σ2K1(m2/2σ2)exp

(−EkEp − kp

2σ2

)def= φG(k, p), (5)

where K1(t) is the modified Bessel function of the 3rd kind of order 1. .. More details

then one can find:

ψ(p, x) = K1(ζm2/2σ2)

ζK1(m2/2σ2)

def= ψG(p, x), with ζ =

√1−

4σ2

m2[σ2x2 + i(px)]

which in the range σ2(x0?)2 m2/σ2, σ2|x?|2 m2/σ2 reads:

ψG(p, x) = exp(imx0? − σ2x2?

)= exp

i(px)− σ2

m2

[(px)2 − m2x2]. (6)

|ψG(p, x)| is invariant under the transformationsx0 7−→ x0 + τ, x 7−→ x+ vpτ... More details

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S-matrix amplitude with wave-packets

Initial and final states read|i〉 = |p1, x1, s1 . . .pN, xN, sN〉, |f〉 = |k1, y1, r1 . . .kn, yn, rn〉

Dimensionless amplitude

A =〈f|S|i〉√〈i|i〉√〈f|f〉

Microscopic probabilityP(pi, xi, si; kf, yf, rf) ≡ |A|2

Macroscopically averaged probability or number of events

dN =∑si,rf

∏i

dpidxi(2π)3

∏f

dpfdyf(2π)3

P(pi, xi, si; kf, yf, rf)ρ(pi, xi, si; kf, yf, rf)

with ρ being the particles density distribution

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An example. The cross-sectionCollision of two wavepackets with impact vector b = x2 − x1 andn = v12/|v12|Microscopic probability reads as ratio of two cross-sections:

P(b,n) = dσS12 e−π|b×n|2/S12 ,

with S12 = S1 + S2 and Si = π/2σ2i and σ is the usual cross-section with

plane waves

S1 S2

bT

bT

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An example. The cross-section. Macroscopic averaging.

Introduce notation bT = b× nAverage over bT assuming the flux does not depend on it:

dN = 〈P(b,n)〉 =∫dbTΦ dσS12 e

−πb2T/S12 = Φdσ.

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Evolution of wave packet

A numerical example: σ2/m2 = 10−10, γ = 105

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Dispersion..

Dispersion of wave-packet lead to a spherical like wave at large times relative to”dispersion time” m/σ2

The wave packet appears as a stable configuration over large times if it isultra-relativistic

Particle σmax, eV Γ/σmax dmin? , cmµ± 1.78× 10−1 1.68× 10−9 1.72× 10−4

τ± 2.01× 103 1.13× 10−6 1.53× 10−8

π± 1.88 1.35× 10−8 1.63× 10−5

π0 3.25× 104 2.41× 10−4 0.94× 10−9

K± 5.12 1.04× 10−8 5.99× 10−6

K0S 6.05× 101 1.22× 10−7 5.07× 10−7

K0L 2.53 5.08× 10−9 1.21× 10−5

D± 1.09× 103 5.82× 10−7 2.82× 10−8

D0 1.73× 103 9.28× 10−7 1.77× 10−8

D±s 1.61× 103 8.18× 10−7 1.91× 10−8

B± 1.46× 103 2.76× 10−7 2.11× 10−8

B0 1.51× 103 2.86× 10−7 2.03× 10−8

B0s 1.55× 103 2.89× 10−7 1.98× 10−8

n 2.64× 10−5 2.81× 10−14 1.16Λ 5.28× 101 4.74× 10−7 5.81× 10−7

Λ±c 2.74× 103 1.87× 10−6 1.12× 10−8

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachFormulation of the problem

Calculation of macroscopic diagram with a block Xs ofthe ”source” and a block Xd of the ”detector”. Theblocks Xs and Xd are separated by macroscopically bigspace-time interval.The source emits neutrino with a time interval τs:

τs → ∞ corresponds to stationary source (Sun,reactor, atmosphere, ...)τs ∼ µs, ns— accelerator neutrino (T2K, Nova,OPERA, MINOS, K2K)

The detector detects neutrino during a time interval τd:

τd τs corresponds to stationary source (Sun,reactor, atmosphere, ...)τd & τs — accelerator neutrino (T2K, Nova,OPERA, MINOS, K2K)

All external particles (incoming or outgoing) arerelativistic wave packetsVirtual neutrino is described by causial propagator

Xd

Xs

Is

i

Fs

Id Fd

Feynman rules: e−ipa(xa−x)ψa (pa, xa − x)

e+ipb(xb−x)ψ∗b (pb, xb − x),

where ψκ (pκ , x) (κ = a, b) wavepackets for every particle. Internallines and loops remain unchanged.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachScheme of calculation of macroscopic amplitude

Normalized amplitude is given by 4th order in perturbation theory in EWcoupling constand g (strong and electromagnetic interactions are taken intoaccount exactly — without perturbations!):

Aβα= 〈out|S|in〉 (〈in|in〉〈out|out〉)−1/2

=1

N

(−ig2√2

)4

〈Fs⊕Fd|T∫dxdx′dydy′ : j`(x)W(x) : : jq(x′)W†(x′) :

× : j†`(y)W†(y) : : j†q(y′)W(y′) : Sh|Is⊕Id〉.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachScheme of calculation of macroscopic amplitude

The hadronic parts of the amplitude can be factorized into a product of sourceand detector subblocksThe ∫ dq in the neutrino propagator can be taking with help of Grimus-Stockinger(GS) theorem giving rise to ∝ e

−iq0T+i√q20−m2j L

L with T = X0d − X0s , L = Xd − XsThe remaining integration over ∫ dq0 can be taken with help of the saddle-pointapproximationAssuming mj = 0 in the matrix elements (not in the exponentials!) the totalamplitude can be factorized into:

.. More details

Aβα =D|Vs(pν)Vd(pν)|MsM∗

di(2π)3/2NL

∑jV∗αjVβj e−Ωj(T,L)−iΘj . (7)

wherethe matrix elements: Ms corresponds to reaction Is → F′s + `+α + ν,

Md corresponds to reaction ν + Id → F′d + `−β .

the phaseΩj(T, L) = i(pjX) +

(2D2j /E2ν

) [(pjX)2 − m2

j X2], X = Xd − Xs.

gauss dispersion of the neutrino energy D is a function of all σ

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachNeutrino wave packet

Given the factorized amplitude it turnes out that we calculated the amplitudeof neutrino propagation (which coincides with its wave-function up to afactor √2Eν) of the outgoing neutrino ψ∗

j = e−Ωj with the explicitlyinvariant complex phase Ωj(T, L) :

Ωj(T, L) = i(pjX) +(2D2

j /E2ν) [

(pjX)2 −m2j X2

], X = Xd − Xs. (8)

the ν WP is of CRGP form with the “width“Σj =

√2Dj/Γj (Γj = Eν/mj).

Since Σj is a complex-valued function, the νWPs spreads with increaseof L = |X|. The spread effect can be important only at “cosmological”distances. Here we limit ourselves to the “terrestrial” conditionsThe relative energy-momentum uncertainty of the νWP isδEj/Ej ∼ δPj/Pj ∼ D/Eν ≪ 1. The mean position of the νWPs evolvesalong the “classical trajectory” L = vjT

.. More details

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachMicroscopic probability.

Recall the dimensionless amplitude


di(2π)3/2NL

∑jV∗αjVβj e−Ωj(T,L)−iΘ. (9)

yielding the microscopic probability:

|Aβα|2 =(2π)4δs(pν − qs)Vs|Ms|2∏

κ∈S 2EκVκ(2π)4δd(pν + qd)Vd|Md|2∏

κ∈D 2EκVκ

× D2

(2π)3L2∣∣∣∑jV∗αjVβj e−Ωj(T,L)

∣∣∣2. (10)

withδs,d - “smeared“ δ functionVs,d - 4-D overlap volume:

Vs,d =∫dx

∏κ∈S,D

|ψκ (pκ , xκ − x)|2 =π2

4|<s,d|−1/2 exp (−2Ss,d). (11)

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachMacroscopic averaging.

The microscopic probability depends on all individual mean coordinates ofthe wave packets, it should be averaged over ensembles in the source anddetector to get the macroscopic probability or the number of events.We assume

particles distribution functions independent on time within timeintervals τs = x02 − x01 and τd = y02 − y01 (could be astronomically large):

fa(pa, sa; x) = θ(x0 − x01) θ (x02 − x0) fa(pa, sa;x) for a∈Is,

fa(pa, sa; y) = θ(y0 − y01) θ (y02 − y0) fa(pa, sa; y) for a∈Id.

source and detectors have finite 3D volumes VS ,VDThen:

〈〈|Aβα|2〉〉 ≡ dNαβ =

∫ ∏κ∈Is,Id,Fs,Fd

dxκdpκfκ(pκ , sκ , xκ)(2π)32EκVκ 〈|Aβα|2〉

=τdVDVS

∫dx

∫dy

∫dΦν

∫dσνDPαβ(Eν , |y− x|).

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachMacroscopic averaging.

〈〈|Aβα|2〉〉 ≡ dNαβ =

∫ ∏κ∈Is,Id,Fs,Fd

dxκdpκfκ(pκ , sκ , xκ)(2π)32EκVκ 〈|Aβα|2〉

=τdVDVS

∫dx

∫dy

∫dΦν

∫dσνDPαβ(Eν , |y− x|).

whereflux density of neutrinos in D, produced through the processesIs → F′s`+αν in S reads: dx

VS∫ dΦν

dEν

the differential cross section of the neutrino scattering off the detectoras a whole.

1

VD∫dydσνD

.. More details

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachQuantum mechanical formula

. Main result of our work is the most general formula for ”oscillationprobability”:

Pαβ(Eν , L) =∑ijV∗αiVαjVβiV∗βjSij exp

(i2πLLij − A 2

ij

). (12)

• This probability coincides with QM formula if Sij = 1 and Aij = 0.Violation of these conditions limits region of applicability of standardformula. One needs to satisfy two conflicting requirements on neutrinoenergy dispersion δEν = D

coherence condition at production (could be violated):

δEν πn

2Lij

destruction of coherence (will be violated at large distances):

δEν Eν Lij2πL

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachQuantum mechanical formula

The Sij is a quite complicated object:

Sij = exp(−B2

ij)

4τdD

2∑l,l′=1

(−1)l+l′+1Ierf

[2D

(x0l − y0l′ +

Lvij

)− iBij

], (13)

Aij = (vj − vi)DL = 2πDLEνLij , Bij =

∆Eji4D

=πn

2DLij , (14)

ϕij =2πLLij , Lij = 4πEν

∆m2ij,

1

vij =1

2

(1

vi +1

vj),

∆m2ij = m2

i −m2j , ∆Eij = Ei − Ej,

Ierf(z) =∫ z

0

dz′erf(z′) + 1√π

= z erf(z) + 1√πe−z2 ,

which depends on:ν energy-momentum uncertainty D, its mean energy Eν , masses mi,mj,correction to energy and momentum n

time intervals of the source τs and the detector τdlength between the source and the detector L and time differencebetween two time windows in the source and the detector T. There is noreason apriori that L ≈ T, these are two independent parameters

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachAsymptotics of a stationary source

In the asymptotic regime t = τsD → ∞ (valid for atmosperic, solar, reactor ν):S(t, t′, b) ∼ exp(−b2)

the “probability“ (12) takes on the form already known from the literature,1

Pαβ(Eν , L) =∑ijV∗αiVαjVβiV∗βj exp

(i2πLLij

− A 2ij − B2

ij

),

=∑ijV∗αiVαjVβiV∗βj exp

i2πLLij −(2πDLEνLij

)2

−(

πn

2DLij

)2

=∑ijV∗αiVαjVβiV∗βj exp

i2πLLij −(LLcohij

)2

−(

1

DLintij

)2

with“coherence length“ Lcohij = 1

∆vijD (∆vij = |vj − vi|),

“interference length“ Lintij = 14∆Eij =

2Lijπn.

1See. e.g., C. Giunti C and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics (OxfordUniversity Press Inc., New York, 2007); M. Beuthe, Oscillations of neutrinos and mesons in quantumfield theory, Phys. Rept. 375 (2003) 105 [arXiv:hep-ph/0109119]; M. Beuthe, Towards a uniqueformula for neutrino oscillations in vacuum, Phys. Rev. D 66 (2002) 013003 [arXiv:hep-ph/0202068].

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachAsymptotics of a stationary source

• The factors exp(−A 2

ij)(with i6=j) suppress the interference terms at the distances

exceeding the “coherence length”

Lcohij =1

∆vijD |Lij| (∆vij = |vj − vi|),

when the νWPs ψ∗i and ψ∗

j are strongly separated in space and do not interfereanymore. Clearly Lcohij → ∞ in the plane-wave limit.• The suppression factors exp

(−B2

ij)(i6=j) work in the opposite situation, when the

external packets in S or D (or in both S and D) are strongly delocalizedThe gross dimension of the the neutrino production and absorption regions in S and Dis of the order of 1/D. The interference terms vanish if this scale is large compared tothe “interference length”

Lintij =1

4∆Eij=

2Lijπn

.

In other words, the QFT approach predicts vanishing of neutrino oscillations in theplane-wave limit. In this limit, the flavor transition probability does not depend on L,Eν , and neutrino masses mi and becomes

PPWLαβ =∑i

|Vαi|2|Vβi|2 ≤ 1.

Thereby, a nontrivial interference of the diagrams with the intermediate neutrinos ofdifferent masses is only possible if D 6= 0.

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachAsynchronized and synchronized measurements

Current accelerator experiments detect neutrinos within tiny time windows:T2K OPERA MINOS MiniBooNe

Nova SciBooNeτs 50ns 10.5µs 10µs 4nsτd 500ns 20µs 100µs #ns

Our formula predicts vanishing of the number of detected events if the timewindows are not synchronized.Let us consider further thecase of “synchronized” mea-surements, in whichx01,2 = ∓τs

2, y01,2 = L∓ τd

2.

τ /2s−τ /2s 0 x 0

L+τ /2dL−τ /2d L y 0

≃T L−

− − −

Thus the factor Sij can be expressed through a real-valued function S(t, t′, b)of three dimensionless variables:

Sij = S (Dτs,Dτd,Bij),2t′S(t, t′, b) = exp (−b2)Re [Ierf (t+ t′ + ib)− Ierf (t− t′ + ib)].

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The diagonal decoherence function 0 ≤ Sii = S (Dτs,Dτd, 0) ≤ 1suppresses the total event rateThe non-diagonal decoherence functions 0 ≤ Sij = S (Dτs,Dτd,Bij) ≤ 1[i 6= j] suppress the interference (“oscillations“)

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The diagonal decoherence function

S(t, t, 0) → 0 for τs τν ∼ 1/D. This is natural to obtain zero events measuringmuch shorter times than the neutrino wave packet time widthS(t, t, 0) → 1 for τs τν ∼ 1/D. No suppression is exected for measurementtime intervals much longer than the neutrino wave packet time width

Note however that the effect of finite the neutrino wave packet time widthτν persists up to τs/τν ∼ 10− 100. This provides a hint to measure neutrinowave packet time width!Another way to measure τν is measure the event rate as a function of τd atfixed τs

.. The details

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Nondiagonal decoherence functionThe decoherence function S(t, t′, b) at b 6= 0 is much more involved.

At very large t, the function S(t, t, b) becomes nearly independent on t, slowlyapproaching the asymptotic behavior S(t, t, b) ∼ exp(−b2) (t, t′ → ∞)... The details

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Nondiagonal decoherence functionAt finite τs, τd the suppression is smaller than the asymptotic limitlimt→∞ S(t, t, b) ∼ exp(−b2). Why?The reason is that a measurement at finite time intervals introduces anadditional uncertainty into energy-momentum thus making the coherence ofstates more probable

.. The details

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Where one can observe a difference between QM and QFT approaches?At accelerators one can expect:

decrease in number of events in near detector due to rather small timeintervals τs, τdmodification of ”oscillation” signal in far detectorHowever, taking into account of found corrections might get ∆m2, sin2 2θright

Intense (Mega-Curie) sources of (anti)neutrino with narrow energy linesδEν Eν mostl likely will be incoherent according to QFT and coherentin QM theory.Cosmological neutrinos are incoherent in QFT based approachLight and Heavy (but still ultra-relativistic) neutrinos will not show an”oscillotary” behaviour as they produced inchorently in QFT while theywill ”oscillate” in QM theoryOur approach gives a completely different picture about sterileneutrinos compared to QM

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Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryNeutrino oscillations in our approachSterile neutrinos

Sterile neutrino — is a popular explanation of experimental ”anomalies”(new degree of freedom — an additional ∆m2)

LSND anomalyMiniBooNE anomalyreactor anomaly

. Basic question: What it is «sterile neutrino»?

. Traditional answer: A state of neutrino which does not interact but astate into which active=interacting neutrinos could ”oscillate”

. However the effect of ”oscillations” is NOT due to mutualtransformation of neutrinos into each other, rather it is a result ofinterference of diagrams with interacting neutrinos.

Could sterile neutrinos exist in QFT?

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In SM there are ”sterile” states — right neutrinos νR. However they aresterile until neutrino is considered massless. Non zero neutrino mass meansthat νL, νR interact with Higgs field. There are two scenarios:1 Number of fields νR equals three: diagonalization λ``′ ν`Lφν`′R + h.c.gives three massive fields of neutrinos. No sterile neutrinos.

2 Number of fields νR is larger than three. Then one has to diagonalize themost general mass lagrangian:

Lm = −1

2

(νL, (νR)c

)( mL mDmD mR

)((νL)c

νR

)+ h.c.

where ν fL = (νeL, νµL, ντL)T, (νR)c = ((νeR)c, (νµR)c, (ντR)c, . . . )T .

Lm is diagonal in terms of new fields νmL ,NL:(νfL

(νR)c

)=

( V MK U

)(νmLNL

)The fields νmL ,NL interact with W, Z— not really sterile!

. . . . . .


The unitarity of mixing matrix gives:VV† +MM† = 1, KK† + UU† = 1, VK† +MU† = 0,

however VV† 6= 1,UU† 6= 1!.Existence of additional neutrino fields lead to a statement that PMNS mixingmatrix could not non unitary!

Interaction with W:LSM/EWcc = − g√

2

∑α=e,µ,τ

`αL γµναL Wµ + h.c.→

− g√2

∑α=e,µ,τ

3∑i=1

Vαi`αL γµνmiLWµ − g√2

∑α=e,µ,τ

3∑i=1

Mαi`αL γµNiLWµ + h.c.

Interaction with Z non diagonal:

LSM/EWnc = − gcos θW

∑`=e,µ,τ

ν`LγµνLZµ →

− gcos θW

(V†VνmL γµνmL +M†MNLγµNL + V†MνmL γµNL +M†VNLγµνmL

)Zµ

. . . . . .


There are at least two data sets which puts limits:LEP from decays Z→ invisible limits M†M 1.Tritium decays also limits me = ∑3

i=1 |Vei|2mi +∑k |Mek|2mk

One can think about possible scenaries:If mN mν , then no oscillations with ∆m2ster could be expected becauseof coherence suppression. The effect can be seen in mixing matrixnon-unitarity and as a result fewer number of events.If mN ' mν , the interference is possible while the contribution from∆m2ster will be suppressed due to smallness of matrix M. It is hard to theinterference (possible only at comparable neutrino masses) + smallcontributon to Z boson width + small contribution to mass me + smallneutrino mass (cosmology) simultaneosly. This makes our approachvery different from QM (thus our theory is falsifiable)

In a consistent QFT based theory of neutrino oscillations the expected effectof ”sterile” neutrino is significantly different from QM approach.

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryConclusions

Within QFT and relativistic wave functions of external states we considered amacroscopic process with lepton number violation. As a result one can find

• calculated neutrino wave function - which turned out to be (in general)a spreading with time-distance wave packet.

• the number of events of macroscopic process is factorized into a productΦ× Pαβ × σ

• The standard QM ν-oscillation formula is an approximation of the moregeneral formula which depends essentialy on. the source “machine ” and detector exposure time;. reaction types in the neutrino production and absorption regions andphase-space domains of these reactions;

. dimensions of the source and detector and distance between them.• The QFT formula as a by product explains a lost of coherence forcharged leptons

• We argue that the neutrino wave function can be measured inexperiments with short beam pulses (T2K, OPERA, MINOS, MiniBooNe,SciBooNe, Nova, ...)

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theory

Appendices. Additional information

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on Wave packetsRelativistic Gaussian packet in momentum space

We consider a simple model of the state – relativistic Gaussian packet (RGP)with:

φ(k, p) = 2π2

σ2K1(m2/2σ2)exp

(−EkEp − kp

2σ2

)def= φG(k, p), (15)

where K1(t) is the modified Bessel function of the 3rd kind of order 1. Inwhat follows we assume

σ2 m2 (16)Then the function (15) can be rewritten as an asymptotic expansion:

φG(k, p) = 2π3/2

σ2

mσexp

[(k− p)24σ2

] [1 +

3σ2

4m2+O

(σ4

m4

)]. (17)

The nonrelativistic limit of the function (17) coincides, up to a normalizationfactor, with the usual (noncovariant) Gaussian distribution:

ϕG(k− p) ∝ exp[− (k− p)2

4σ2

].

But it is not the case at relativistic and especially ultrarelativistic momenta... Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on Wave packetsRelativistic Gaussian packet in momentum space

Example: in the vicinity of the maximum k = p

φG(k, p) ≈ 2π3/2

σ2

mσexp

[− (k− p)2

4σ2Γ 2p

](k ∼ p).

We see that in this case the relativistic effect consists in a “renormalization”of the WP width (σ → σΓp). This renormalization is very essential for theneutrino production and detection processes involving relativistic particles... Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on Wave packetsRelativistic Gaussian packet in coordinate space

Function ψG(p, x).

x /m2 0

| |G

x /m2 3

A 3D plot of |ψG(0, x?)| as a functionof σ2x0?/m and σ2x3?/m (assumingthat x? = (0, 0, x3?)). The calculationsare done for σ/m = 0.1.

ψ(p, x) = K1(ζm2/2σ2)

ζK1(m2/2σ2)

def= ψG(p, x),

where we have defined the dimensionlessLorentz-invariant complex variable

ζ =

√1−

4σ2

m2[σ2x2 + i(px)];

|ζ|4 =

[1−

4σ4x2m2

]2+

16σ4(px)2m4

,

ϕ = −1

2arcsin

[4σ2(px)m2|ζ|2

].

It can be proved that for any p and x|ζ| ≥ 1 and |ϕ| < π/2.

.. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on Wave packetsRelativistic Gaussian packet in coordinate space

An analysis of the asymptotic expansion of ln [ψG(0, x?)] in powers of σ2/(m2ζ)provides the following (necessary and sufficient) conditions of the nondiffluentbehavior:

σ2(x0?)2 m2/σ2, σ2|x?|2 m2/σ2. (18a)They can be rewritten in the equivalent but explicitly Lorentz-invariant form:

(px)2 m4/σ4, (px)2 − m2x2 m4/σ4. (18b)Under these conditions ψG(p, x) reduces to the simple form:

ψG(p, x) = exp(imx0? − σ2x2?

)= exp

i(px)− σ2

m2

[(px)2 − m2x2]. (19)

Some properties of CRGP:1. The mean coordinate of the packet follows the classical trajectory (CT) x = vpx0.2. |ψG(p, x)| = 1 along the CT and |ψG(p, x)| < 1 with any deviation from it.3. |ψG(p, x)| is invariant under the transformations x0 7−→ x0 + τ, x 7−→ x+ vpτ.4. In the nonrelativistic limit, the wave function (19) takes the form:

ψG(p, x) ≈ exp[im (x0 − vpx)− σ2 |x− vpx0|2

].

.. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the amplitude of the macroscopic processFeynman rules and overlap integrals.

The additional factors (1) provide the following two common multipliers in theintegrand of the scattering amplitude [we will call these the overlap integrals]:

Vs(q) =∫dxe+iqx

[∏a∈Ise−ipaxaψa (pa, xa − x)

][ ∏b∈Fseipbxbψ∗

b (pb, xb − x)],

Vd(q) =∫dxe−iqx

[ ∏a∈Ide−ipaxaψa (pa, xa − x)

][ ∏b∈Fdeipbxbψ∗

b (pb, xb − x)].(20)

The function Vs (Vd) characterizes the 4D overlap of the “in” and “out” wave-packetstates in the source (detector) vertex .. Return

In the plane-wave limit (σκ → 0, ∀κ)Vs(q) → (2π)

4δ (q− qs) and Vd(q) → (2π)

4δ (q+ qd),

where qs and qd are the 4-momentum transfers defined byqs =

∑a∈Ispa −

∑b∈Fspb and qd =

∑a∈Idpa −

∑b∈Fd

pb.

The δ functions provide the energy-momentum conservation in the vertices s and d (thatis in the “subprocesses” Is → Fs + ν∗

i and ν∗i + Id → Fd) and, as a result, – in the whole

process:Is⊕Id → Fs⊕Fd :

∑a∈Is⊕Id

pa =∑

b∈Fs⊕Fdpb.

Information about the space-time coordinates of the interacting packets is completely lost.

. . . . . .


In the general case of nonzero spreads σκ , we may expect no more than anapproximate conservation of energy and momentum and lack of any singularities.To quantify this expectation, we apply the CRGP model.

Tµνκ = σ2κ (uµκuνκ − gµν), [uκ = pκ/mκ = Γκ(1, vκ)]⇓

Vs,d(q) =∫dx exp

[i(± qx− qs,dx

)−∑

κ∈S,DTµνκ (xκ − x)µ (xκ − x)ν

],

where S = Is⊕Fs and D = Id⊕Fd. It is useful to define also the tensors<µνs =

∑κ∈STµνκ and <µν

d =∑κ∈DTµνκ .

A crucial property:Tµνκ xµxν = σ2

κ[(uκx)2 − x2] = σ2

κx2? ≥ 0 =⇒ <µνs,d xµxν > 0.

Consequently, there exist the positive-definite tensors <µνs and <µν

d such that<s,d = ||<µν

s,d || = g<−1s,d g, <s,d = ||<µν

s,d ||.

The explicit form and properties of these tensors and relevant convolutions areestablished (studied in detail for the most important reactions1 → 2, 1 → 3, 2 → 2). .. Return

. . . . . .


The overlap integrals In CRGP are the 4D Gaussian integrals in Minkowski space.

Vs,d(q) = (2π)4δs,d(q∓qs,d

)exp

[−Ss,d ± i

(q∓qs,d

)·Xs,d

],

δs,d(K) = (4π)−2|<s,d|−1/2 exp(−1

4<µνs,dKµKν

),

Ss,d =∑κ,κ′

(δκκ′Tµν

κ − Tµκµ′ <µ′ν′

s,d Tνκ′ν′

)xκµxκ′ν ,

Xµs,d = <µνs,d∑κTλκνxκλ.

Physical meaning of δs,d, Ss,d, and Xs,d.• From the integral representation δs,d(K) =

∫ dx(2π)4

exp(−<µν

s,d xµxν + iKx)it

follows that δs,d(K) → δ(K) in the plane-wave limit [σκ → 0, ∀κ ⇒ <µνs,d → 0].

=⇒ just the factors δs (q− qs) and δd (q+ qd) are responsible for theapproximate energy–momentum conservation (with the accuracy governed bythe momentum spreads of the interacting packets) in the neutrino production anddetection points.

.. Return

. . . . . .


• The functions exp (−Ss) and exp (−Sd) are the geometric suppression factorsconditioned by a partial overlap of the in and out WPs in the space-time regionsof their interaction in the source and detector.

This can be seen after converting Ss,d to the form2

Ss,d =∑κTµνκ(xκ − Xs,d

)µ

(xκ − Xs,d)ν

(21)

and taking into account that both Ss,d and Xs,d are invariants under the group ofuniform rectilinear motions (here, τκ are arbitrary real time parameters)x0κ 7−→ x0κ = x0κ + τκ , xκ 7−→ xκ = xκ + vκτκ

Due to this symmetry, (21) can be rewritten asSs,d =

∑κσ2κ

[(Γ 2κ − 1

) (b0κ)2 + b2κ]=∑κσ2κ |b?κ |2,

b0κ=(x0κ − X0s,d

)− |vκ |−1nκ ·

(xκ − Xs,d),

bκ=(xκ − Xs,d

)−[nκ ·

(xκ − Xs,d)] nκ ,

[nκ =

vκ/|vκ |, for vκ 6= 0,

0, for vκ = 0.

]The 4-vector bκ = (b0κ , bκ) is a relativistic analog of the usual impact parameter, so itis natural to call it the impact vector. [Note that |bκ | =

∣∣nκ ×(xκ − Xs,d

)∣∣ forvκ 6= 0.]The 4-vectors Xs,d can be called, accordingly, the impact points. .. Return2In this derivation we have used the translation invariance of the functions Ss,d.

. . . . . .


The suppression of the overlap integral caused by the factors exp(−Ss) and exp(−Sd)can be small only if all the world tubes in the source/detector inter-cross each other.

X

xa

xb

xc

~

~

~

time

space

Artistic view of the “classical world tubes” of interacting wave packets. The tubes reproducethe space-time cylindrical volumes swept by the classically moving spheroids which representthe wave-packet patterns. The impact point X is defined by the velocities vκ of the packetsand the space-time points xκ = xκ(τκ) arbitrarily chosen on the axes of the tubes. The axesdo not necessarily cross the point X.

The interacting packets behave, bluntly speaking, like colliding interpenetrativecloudlets. .. Return .

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the amplitude of the macroscopic processCalculating the amplitude

1. Quark-lepton blocks. We use the Standard Model (SM) phenomenologicallyextended by inclusion of a neutrino mass term. The quark-lepton blocks are describedby the Lagrangian

LW(x) = −g

2√2

[j`(x)W(x) + jq(x)W(x) + H.c.

],

where g is the SU(2) coupling constant, j` and jq are the weak charged currents:jµ` (x) =

∑αiV∗αi ν i(x)Oµ`α(x), jµq (x) =

∑qq′V′∗qq′ q(x)Oµq′(x), [Oµ = γµ(1− γ5)] .

Here Vαi (α = e, µ, τ ; i = 1, 2, 3) and V′qq′ (q = u, c, t; q′ = d, s, b) are the elements ofthe neutrino and quark mixing matrices (V and V′, respectively).The normalized amplitude is given by the 4th order of the perturbation theory in g:

Aβα= 〈out|S|in〉 (〈in|in〉〈out|out〉)−1/2

=1

N

(−ig2√2

)4

〈Fs⊕Fd|T∫dxdx′dydy′ : j`(x)W(x) : : jq(x′)W†(x′) :

× : j†`(y)W†(y) : : j†q(y′)W(y′) : Sh|Is⊕Id〉. (22)The normalization factor N in the CRGP approximation is given by

N 2 = 〈in|in〉〈out|out〉 =∏

κ∈Is⊕Id⊕Fs⊕Fd2EκVκ(pκ).

.. Return

. . . . . .


2. Hadronic blocks. The strong and (possibly) electromagnetic interactionsresponsible for nonperturbative processes of fragmentation and hadronization aredescribed by the hadronic (QCD) interaction Lagrangian Lh(x) and the correspondingpart of the full S-matrix is

Sh = exp[i∫dzLh(z)

].

The following factorization theorem can be proved〈F′s⊕F′d|T

[: jµq (x) : Sh : j†νq (y) :

]|Is⊕Id〉 = J µ

s (pS)J ν†d (pD)

×[∏a∈Ise−ipaxaψa(pa, xa − x)

][ ∏b∈F′seipbxbψ∗

b (pb, xb − x)]

×[ ∏a∈Ide−ipaxaψa(pa, xa − y)

][ ∏b∈F′deipbxbψ∗

b (pb, xb − y)].

Here Js(pS) and Jd(pD) are the c-number hadronic currents in which the stronginteractions are taken into account nonperturbatively, and pS and pD denote the sets ofthe momentum and spin variables of the hadronic states.

The proof is based on the assumed narrowness of the WPs in the momentum space, macro-scopic remoteness of the interaction regions in the source and detector vertices, and theconsideration of translation invariance.

The explicit form of the hadronic currents Js and Jd is not needed for our purposes... Return

. . . . . .


Now, by applying Wick’s theorem, factorization theorem, and the known properties ofthe leptonic WPs, the amplitude (22) can be rewritten in the following way:

Aβα =g4

64N∑jVβjJ ν†

d u(pβ)Oν′Gjν′µ′

νµ (pκ , xκ)Oµ′v(pα)J µs V∗αj, (23)

Gjν′µ′

νµ (pκ , xκ) =∫ dq

(2π)4Vd(q)∆ν′

ν (q− pβ)∆j(q)∆µ′µ (q+ pα)Vs(q). (24)

Here Vs(q) and Vd(q) are the overlap integrals; ∆j and ∆νµ are the propagators of,

respectively, the massive neutrino νj and W boson.3∆j(q) = i (q− mj + i0)−1

.

3. Large-distance asymptotics. At large values of |Xs − Xd|, the integral (24) can beevaluated by means of the Grimus-Stockinger (GS) theorem.4Let F(q) be a thrice continuously differentiable function such that F itself and its 1st and2nd derivatives decrease not slowly than |q|−2 as |q| → ∞. Then in the asymptotic limit ofL = |L| → ∞,∫ dq

(2π)3F(q)eiqLs− q2 + i0 ∼

− 1

4πL F(√s L/L) exp (i√sL)+ O

(L−3/2

)at s > 0,

O(L−2

) at s < 0.

The integrand in (24) satisfies the formulated requirements... Return3The explicit form of ∆µν is not used below. So ∆µν can be thought of as renormalizedpropagator.4W. Grimus and P. Stockinger, Real oscillations of virtual neutrinos, Phys. Rev. D 54 (1996) 3414

[arXiv:hep-ph/9603430].

. . . . . .


4. Integration in q0. The integral over q0, which remains after applying the GStheorem can be evaluated by the regular saddle-point method.In the ultrarelativistic approximation (q0s ≈ −q0d mj, j = 1, 2, 3) the stationarysaddle point q0 = Ej can be found as a series in powers of the small parameterrj = m2

j /(2E2ν). We obtainq0 ≡ Ej = Eν

[1− nrj − mr2j +O

(r3j)],

|qj|q0=Ej ≡ Pj = Eν[1− (n+ 1)rj −

(n+ m+

1

2

)r2j +O

(r3j)],

PiEj

≡ vj = 1− rj −(2n+

1

2

)r2j +O

(r3j), E2j − P2j = m2

j ;

n =YlYl , m = n

(3

2+ 2n

)+

1

R∑

n=1,2,3

(<0ns + <0n

d)ln,

Yµ = <µνs qsν − <µν

d qdν , R =(<µνs + <µν

d)lµlν ,

Eν =YlR , l = (1, l), l = L

L , L = Xd − Xs. .. ReturnThe quantities Ej, Pj = Pjl and vj = vjl can naturally be treated as, respectively, theeffective energy, momentum and velocity of the virtual neutrino νj.

The ultrarelativistic approximation is, of course, reference-frame dependent. That is whythe obtained result is not explicitly Lorentz-invariant.

. . . . . .


In the limit of mj = 0 and assuming the exact energy-momentum conservation,Ej = Pj = Eν = q0s = −q0d .

But, in the general case, the effective neutrino 4-momentum pj = (Ej, pj) is determinedby the mean momenta and momentum spreads of the external WPs involved in theprocess.

Below, we’ll limit ourselves to the 1st order of the expansion in rj. However, the next-order corrections are needed to define properly the range of applicability of the obtainedresult.

Finally, by introducing the notationΩj(T, L) = i

(EjT− PjL)+ 2(Dj/Pj

)2 (PjT− EjL)2,Θ = Xsqs + Xdqd, L = |Xd − Xs|, T = X0d − X0s ,

Dj = Dj

(1 +

8irjE2νD2j L

P3j

)−1/2

, Dj =1 + nrj√

2R , .. Return

we arrive at the saddle-point estimate of the function (24):

Gjν′µ′

νµ = ∆ν′ν (pj − pβ)(pj + mj)∆µ′

µ (pj + pα)|Vd(pj)Vs(pj)|Dje−Ωj(T,L)−iΘ

i(2π)3/2L . (25)This formula can be (and must be) somewhat simplified by putting rj = 0 everywherewherever it is not a factor multiplying L or T (whose values can be arbitrary large).Then the 4-vector pj is replaced by the light-cone 4-vector pν = (Eν , pν) = Eν l.

. . . . . .


Excursus: Neutrino wave packet (νWP).The complex phase Ωj(T, L) can be written in an explicitly invariant form:

Ωj(T, L) = i(pjX) +(2D2j /E2ν

) [(pjX)2 − m2

j X2], X = Xd − Xs. (26)

Comparing the factor ψ∗j = e−Ωj in (25) with the generic CRGP wave function (19),

we conclude that ψ∗j can be treated as the (outgoing) νWPs in which the role of the

parameter σ is played by the functionΣj =

√2Dj/Γj, (Γj = Eν/mj).

Since Σj is a complex-valued function, the νWPs spreads with increase of L = |X|.The spread effect can be important only at “cosmological” distances. Here we limitourselves to the “terrestrial” conditions, for which it is a fortiori possible to judge that

EjL(ΓjEj/2Dj

)2.

⇓Dj ' Dj ' 1/

√2R ≡ D and Σj '

√2D/Γj = 1/(Γj

√R).The relative energy-momentum uncertainty of the νWP isδEj/Ej ∼ δPj/Pj ∼ D/Eν ≪ 1. Of course, the mean position of the νWPs evolves alongthe “classical trajectory” L = vjT, the quantum deviations from which, δL, aresuppressed by the factor

exp−2D2

[δL2/Γ 2

j + (lδL)2].

.. Return

. . . . . .


5. Source-detector factorization. Now, by applying the identityP−pνP+ = P−u−(pν)u−(pν)P+,

where u−(pν) is the Dirac bispinor for the left-handed massless neutrino andP± = 1

2(1± γ5), we define the matrix elements

Ms = (g2/8) u−(pν)J µs ∆µ′

µ (pν + pα)Oµ′u(pα),M∗d = (g2/8) v(pβ)Oµ′∆µ′

µ (pν − pβ)J µ†d u−(pν),

which describe the reactions with production and absorption of a real masslessneutrino ν: Ms corresponds to reaction Is → F′s + `+α + ν,

Md corresponds to reaction ν + Id → F′d + `−β .

The final expression for the amplitude (23) is


di(2π)3/2NL

∑jV∗αjVβj e−Ωj(T,L)−iΘ. (27)

• The form of eq. (27) suggests that it is common for essentially any class ofmacrodiagrams, with exchange of virtual neutrinos between the source and detector,until we do not specify the explicit form of the matrix elements Ms and Md.• To obtain similar answer for the macrodiagrams with an exchange of virtualantineutrinos, one has to replace (besides the matrix elements) V 7−→ V†. .. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the probability of the macroscopic processMicroscopic probability.

It can be shown that|Vs,d(pν)|2 = (2π)4δs,d(pν∓qs,d)Vs,d, (28)

where δs,d are the “smeared” δ functions (analogous to the functions δs,d) andVs,d are the effective 4D overlap volumes of the external packets in thesource and detector;

δs,d(K) = (2π)−2|<s,d|−1/2 exp(−1

2<µνs,dKµKν

), (29)

Vs,d =∫dx

∏κ∈S,D

|ψκ (pκ , xκ − x)|2 =π2

4|<s,d|−1/2 exp (−2Ss,d). (30)

⇓

|Aβα|2 =(2π)4δs(pν − qs)Vs|Ms|2∏

κ∈S 2EκVκ(2π)4δd(pν + qd)Vd|Md|2∏

κ∈D 2EκVκ

× D2

(2π)3L2∣∣∣∑jV∗αjVβj e−Ωj(T,L)

∣∣∣2. (31)

This is the microscopic probability dependent on the mean momenta pκ , initialcoordinates xκ , masses mκ , and parameters σκ of all external wave packetsparticipated in the reaction.

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the probability of the macroscopic processMicroscopic probability.

By using the explicit form of the functions δs,d and D, one can prove thefollowing approximate relation:

2√πDδs (pν − qs) δd (pν + qd) F(pν) =

∫dE′νδs

(p′ν − qs)δd

(p′ν + qd) F(p′ν),

(32)where F(pν) is an arbitrary slowly varying function and p′ν = (E′ν , p′ν) = E′ν l.The relation is valid with the same accuracy with which the amplitude (27)itself has been deduced that is, with the accuracy of the saddle-pointmethod. With help of (32) the squared amplitude (31) transforms to

|Aβα|2 =

∫dEν (2π)4δs(pν − qs)Vs|Ms|2∏

κ∈S 2EκVκ

(2π)4δd(pν + qd)Vd|Md|2∏κ∈D 2EκVκ

× D

2√π(2π)3L2

∣∣∣∑jV∗αjVβj e−Ωj(T,L)

∣∣∣2. (33)

The probability (33) is the most general result of this work. However, it istoo general to be directly applied to the contemporary neutrino oscillationexperiments.

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the probability of the macroscopic processMacroscopic averaging.

To obtain the observable quantities, the probability must beaveraged/integrated over all the unmeasurable or unused variables ofincoming/outgoing WP states.A thought experiment:Assume that the statistical distributions of the incoming WPs a ∈ Is,d over themean momenta, spin projections, and space-time coordinates in the sourceand detector “devices” can be described by the one-particle distributionfunctions fa(pa, sa, xa). It is convenient to normalize each function fa to thetotal number, Na(x0a), of the packets a at a time x0a :∑

sa

∫ dxadpa(2π)3

fa(pa, sa, xa) = Na(x0a) (a ∈ Is,d).

For clarity purposes, we (re)define the terms “source” and “detector”:S = suppxa; a∈Is

∏afa(pa, sa, xa), D = suppxa; a∈Id

∏afa(pa, sa, xa).

We’ll use the same terms and notation S and D also for the correspondingdevices. .. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the probability of the macroscopic processMacroscopic averaging...Innocent assumptions:[1] S and D are finite and mutually disjoint within the space domain.[2] Effective spatial dimensions of S and D are small compared to the mean distance betweenthem but very large compared to the effective dimensions (∼ σ−1

κ ) of all WPs in S and D.[3] The experiment measures only the momenta of the secondaries in D and (due to [2]) thebackground events caused by the secondaries falling into D from S can be neglected.

[4] The detection efficiency in D is 100%.With all these assumptions, the macroscopically averaged probability reads

〈〈|Aβα|2〉〉 ≡ dNαβ =∑spins

∫ ∏a∈Is

dxadpafa(pa, sa, xa)(2π)32EaVa

∫ ∏b∈Fs

dxbdpb(2π)32EbVb

Vs

×∫ ∏a∈Id

dxadpafa(pa, sa, xa)(2π)32EaVa

∫ ∏b∈Fd

dxb[dpb](2π)32EbVb

Vd

×∫dEν(2π)4δs(pν − qs)|Ms|2(2π)4δd(pν + qd)|Md|2

×D

2√π(2π)3L2

∣∣∣∑jV∗αjVβj e−Ωj(T,L)

∣∣∣2. (34)

.∑spins denotes the averaging/summation over the spin projections of the in/out states.

. Symbol [dpb] indicates that integration in variable pb is not performed, i.e.,∫[dpb] = dpb.

Clearly, 〈〈|Aβα|2〉〉 represents the total number, dNαβ , of the events recorded in D and consistedof the secondaries b∈Fd having the mean momenta between pb and pb + dpb.

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the probability of the macroscopic processMacroscopic averaging...Under additional assumptions, the unwieldy expression (34) can be simplified in a few steps.Step 1: Multidimensional integration in WP positions.Supposition 5: The distribution functions fa(pa, sa, xa), as well as the factors e−Ωj−Ω∗

i /L2 varyat large (macroscopic) scales.

The integrand∏κ |ψκ (pκ , xκ − x)|2 in the integral representation of the overlap volumes (30) isessentially different from zero only if the classical word lines of all packets κ pass through a small(though not necessarily microscopic) vicinity of the integration variable.

Supposition 6: The edge effects can be neglected (a harmless extension of supposition [2]).As a result, expression (34) is reduced to the following:

dNαβ =∑spins

∫dx∫dy∫dPs

∫dPd

∫dEν

D∣∣∣∑j V∗αjVβj e−Ωj(T,L)

∣∣∣216π7/2|y− x|2 , (35)

where T = y0 − x0, L = |y− x| and we have defined the differential forms

dPs=∏a∈Is

dpafa(pa, sa, x)(2π)32Ea

∏b∈Fs

dpb(2π)32Eb

(2π)4δs(pν − qs)|Ms|2, (36a)

dPd=∏a∈Id

dpafa(pa, sa, y)(2π)32Ea

∏b∈Fd

[dpb](2π)32Eb

(2π)4δd(pν + qd)|Md|2. (36b)

. . . . . .


Step 2: Integration in time variables.Supposition 7: During the experiment, the distribution functions fa in S and D varyslowly enough with time so that they can be modelled by the “rectangular ledges”

fa(pa, sa; x) = θ(x0 − x01

)θ(x02 − x0) fa(pa, sa; x) for a∈Is,

fa(pa, sa; y) = θ(y0 − y01

)θ(y02 − y0) fa(pa, sa; y) for a∈Id. (37)

Supposition 8: The time intervals needed to switch on and switch off the source anddetector are negligibly small in comparison with periods of stationarity τs = x02 − x01and τd = y02 − y01.

In case of detector, the step functions in (37) can be thought as the “hardware” or “soft-ware” trigger conditions. The periods of stationarity τs and τd can be astronomically long,as it is for the solar and atmospheric neutrino experiments (τs ≫ τd in these cases), orvery short, like in the experiments with short-pulsed accelerator beams (when usuallyτs . τd).

Within the model (37), the only time-dependent factor in the integrand of (35) ise−Ωj−Ω∗

i . So the problem is reduced to the (comparatively) simple integral∫ y02y01dy0

∫ x02x01dx0 e−Ωj(y0−x0,L)−Ω∗

i (y0−x0,L) =√π

2Dτd exp

(iϕij − A 2

ij)Sij. (38)

. . . . . .


In this relation, we have adopted the following notation:

Sij =exp

(−B2

ij)

4τdD

2∑l,l′=1

(−1)l+l′+1Ierf

[2D

(x0l − y0l′ +

Lvij

)− iBij

], (39)

Aij = (vj − vi)DL = 2πDLEνLij

, Bij =∆Eji4D

=πn

2DLij, (40)

ϕij =2πLLij

, Lij = 4πEν∆m2

ij,

1

vij=

1

2

(1

vi+

1

vj

),

∆m2ij = m2

i − m2j , ∆Eij = Ei − Ej,

Ierf(z) =∫ z0dz′erf(z′) + 1

√π

= z erf(z) + 1√πe−z2 ,

For a more realistic description of the beam pulse experiments, the model (37) could bereadily extended by inclusion of a series of rectangular ledges followed by pauses duringwhich fa = 0.

Then substituting (38) into (35) we obtain:dNαβ = τd

∑spins

∫dx∫dy∫dPs

∫dPd

∫dEν Pαβ(Eν , |y− x|)

4(2π)3|y− x|2 , (41a)

≡τdVDVS

∫dx∫dy∫dΦν

∫dσνDPαβ(Eν , |y− x|). (41b)

The differential forms dPs,d in (41a) are are given by eq. (36) after substitutionfa 7−→ fa.

. . . . . .


Explanation of the factors in eq. (41b).. VS and VD are the spatial volumes of the source and detector,respectively.. The differential form dΦν is defined in such a way that the integral

dxVS

∫ dΦν

dEν = dx∑spins∈ S

∫ dPsEν2(2π)3|y− x|2 (42)

is the flux density of neutrinos in D, produced through the processesIs → F′s`+αν in S.

More precisely, it is the number of neutrinos appearing per unit time and unit neutrinoenergy in an elementary volume dx around the point x ∈ S, travelling within the solidangle dΩν about the flow direction l = (y− x)/|y− x| and crossing a unit area, placedaround the point y ∈ D and normal to l.

.. Return

. . . . . .


. The differential form dσνD is defined in such a way that1

VD∫dydσνD =

∑spins∈ D

∫ dydPd2Eν (43)

represents the differential cross section of the neutrino scattering off thedetector as a whole.

In the particular (and the most basically important) case of neutrino scattering in the re-action νa→ F′d`−β , provided that the momentum distribution of the target scatterers a issufficiently narrow, the differential form dσνD becomes exactly the elementary differen-tial cross section of this reaction multiplied by the total number of the particles a in D.

.. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the “probability of neutrino oscillations“Diagonal decoherence function

Let us now return to the de-coherence factor, limiting our-selves to a consideration of“synchronized” measurements,in whichx01,2 = ∓τs

2, y01,2 = L∓ τd

2.

τ /2s−τ /2s 0 x 0

L+τ /2dL−τ /2d L y 0

≃T L−

− − −

With certain technical simplifications, the factor (39) can be expressedthrough a real-valued function S(t, t′, b) of three dimensionless variables,namely:

Sij = S (Dτs,Dτd,Bij),2t′S(t, t′, b) = exp (−b2)Re [Ierf (t+ t′ + ib)− Ierf (t− t′ + ib)].

Diagonal decoherence functionS(t, t′, 0) = 1

2t′[Ierf (t+ t′)− Ierf (t− t′)] ≡ S0(t, t′), (44)

This function corresponds to the noninterference (neutrino mass independent)decoherence factors Sii. The following inequalities can be proved:

0 < S0(t, t′) < 1, S0(t, t′) < t/t′ for t′ ≥ t, S0(t+ δt, t) > erf(δt) for δt > 0.

.. Return

. . . . . .


The strong dependence of the common suppression factor S0(t, t′) on its arguments att . t′ provides a potential possibility of an experimental estimation of the function D(or, rather, of its mean values within the phase spaces), based on the measuring thecount rate dRαβ = dNαβ/τd as a function of τd and τs (at fixed L) and comparing thedata with the results of Monte-Carlo simulations.The optimal strategy of such an experiment should be a subject of a dedicatedanalysis. .. Return

. . . . . .


For the important special case, t′ = t (representative, in particular, for the experimentswith accelerator neutrino beams), we find

S0(t, t) = erf(2t)− 1− e−4t2

2√πt ≈

2t√π

(1−

2t23

+8t415

)for t 1,

1−1

2√πt for t 1.

(45)

.. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryDetails on the “probability of neutrino oscillations“Non diagonal decoherence function

S(t, t′, 0.1). S(t, t′, 0.2). S(t, t′, 0.3).

.. Return

. . . . . .


S(t, t′, 0.4). S(t, t′, 0.5). S(t, t′, 0.6).

.. Return

. . . . . .


S(t, t′, 0.7). S(t, t′, 0.8). S(t, t′, 0.9).

.. Return

. . . . . .


S(t, t′, 1.0). S(t, t′, 1.5). S(t, t′, 2.0).

.. Return

. . . . . .


S(t, t′, 3.0). S(t, t′, 4.0). S(t, t′, 5.0).

.. Return

. . . . . .


S(t, t′, 6.0). S(t, t′, 7.0). S(t, t′, 8.0).

.. Return

. . . . . .


S(t, t′, 9.0). S(t, t′, 10.0). S(t, t′, 15.0)/S0(t, t′).

.. Return

. . . . . .


S(t, t′, 0.10)/S0(t, t′),S(t, t′, 0.50)/S0(t, t′).

S(t, t′, 0.75)/S0(t, t′),S(t, t′, 1.00)/S0(t, t′).

S(t, t′, 1.50)/S0(t, t′),S(t, t′, 4.00)/S0(t, t′).

.. Return

. . . . . .

Vacuum neutrino oscillations with relativistic wave packets in quantum field theoryOPERA anomaly

Summary of OPERA results

Advance arrival of νµ from CERN to LNGSδt = 57.8± 7.8(stat.)+8.3

−5.9(sys.)this corresponds to

(v− c)/c = (2.37± 0.32(stat.)+0.34−0.24(sys.))× 10−5

These results were reproduced by a test performed with a beam with ashort-bunch time-structure allowing measuring the neutrino time offlight at the single interaction level.

. . . . . .


Possible explanations (> 140 papers already)

Superluminal ν. Lorentz symmetry violation.Anomalous refraction in Earth interior, dark matter, etcFifth force, new fields, exoticsGR correctionsNew geometry, several fundamental speed limitsextra dimensionsWave packets, off-shell ν, neutrino oscillations

. . . . . .


Our explanationIt is possible that neutrino wave packet (WP) is macroscopically large in thetransverse size being microscopic in the direction of motion. This couldmimic early arrival of neutrino.d⊥

(0.1 eVmν

)km, d‖ =

d⊥Γν

10−2

(10 GeVEν

)(0.1 eVmν

)µm.

DetectorSource

Neutrino

wavepacket

vν

A

B

CDE

θ

r

AB = AD = L

. . . . . .


Relativistic Gauss Wave Packet

We built an explicit example of WP - Relativistic Gauss Wave Packet

ψ(p, x) = K1(ζm2/2σ2)

ζK1(m2/2σ2)

def= ψG(p, x), (46)

with dimensionless variable:

ζ = |ζ|eiϕ =

√1− 4σ2

m2[σ2x2 + i(px)]. (47)

RGP is a solution of Klein-Gordon equationRGP is in general a dispersing with time wave packet. However itsdispersion can be not so fast as one usually assumes.

. . . . . .




. . . . . .




. . . . . .




. . . . . .




. . . . . .




. . . . . .




. . . . . .


For the given example RGP becomes a spherical wave like att ' 104m/σ2. At shorter distances ψG(x) behaves like a stable structure!For realistic parameters of neutrino RGP the spherical asymptoticsbecomes even laterFor CERN-LNGS environment neutrino wave packet can be written as anon -dispersing approximation of ψG(x):

ψ∗ν = eiEν(x0−vνx)−σ2

νΓ2ν (x‖−vνx0)2−σ2

νx2⊥ .

. . . . . .


Probability of neutrino CC interaction at LNGS as a function oftransverse radius r from the beam axis

Distance from beam axis (km)

CC

eve

nts/

(ton

10

PoT

) 13

00 1 2 3 4 5 6

0.6

0.1

0.2

0.3

0.4

0.5

FWHM ≈ 2.8 km

. . . . . .


Calculation of OPERA expected arrival time distribution with andwithout our effect

t (ns)0 2000 4000 6000 8000 10000

0

0.2

0.4

0.6

0.8

t (ns)0 200 400 600 800 1000 1200

0

0.2

0.4

0.6

0.8

t (ns)8500 9000 9500 10000 10500 11000 11500 12000

OPERA PDFThis work

. . . . . .


Some outlook

Without any free parameter we find that OPERA should see some 20 nsearlier arrival of neutrino with similar varianceLeft edge of the time distribution is expected to be shifted by some 60ns, while the right one y some 20-25 ns.Similar calculations performed for MINOS give 120ns earlier arrival ofneutrino with surprizingly good agreement withe MINOS result

δt = (126± 32stat ± 64sys) нс (68% C.L.).

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